## Black-Scholes Model

The Black-Scholes Model is a mathematical formula designed to price an option. It is a function of a number of variables such as stock price, striking price, volatility, time to expiration, dividends to be paid, and the current risk-free interest rate. It is a fairly common method of option pricing but does use some assumptions which give rise to limitations. The main assumptions of the Black-Scholes Model are;

• 1. The underlying Stock pays no dividends for the duration of the option, which may not be true all the time.
• 2. Option can only be exercised upon expiration, which is true for European style options but not American styel options. Most of the exchange traded options are American style, meaning the option can be exercised any time up to the expiry date.
• 3. The market direction cannot be predicted
• 4. No commissions are charged in the transaction
• 5. Interest rates remain constant during the option period, which may not always be true.
• 6. Volatility is constant over the duration of the option.

• Although the Black-Scholes Model has these limitations compared to other more complex option pricing models, it is still popular with option traders due to it's relative simplicity. For those of you who are mathematically inclined the Black-Scholes formula is as follows;

C0 = S0N(d1) - Xe-rTN(d2)
Where:
d1 = [ln(S0/X) + (r + σ2/2)T]/ σ √T
And:
d2 = d1 - σ √T

And where:
C0 = current option value
S0 = current stock price
N(d) = the probability that a random draw from a standard normal distribution will be less than (d).
X = exercise price
e = 2.71828, the base of the natural log function
r = risk-free interest rate (usually the money market rate for a maturity equal to the option's maturity.)
T = time to option's maturity, in years
ln = natural logarithm function
σ = standard deviation of the annualized continuously compounded rate of return on the stock

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